What is the average value of $-x^2+3$ on the interval $[0,6]$ ?
Explanation: In general, this is the average value of function $f$ over the interval $[a,b]$ : $\dfrac{\int_a^b f(x)\,dx}{b-a}$ In our case, ${f(x)=-x^2+3}$, ${a=0}$ and ${b=6}$ : $\begin{aligned} \dfrac{\int_{ a}^{ b} {f(x)}\,dx}{ b- a}&=\dfrac{\int_{{0}}^{ {6}} ({-x^2+3})\,dx}{{6}-{0}} \\\\ &=\dfrac{\Big[-\dfrac{x^3}{3}+3x\Big]_{0}^{6}}{6} \\\\ &=\dfrac{-54-0}{6} \\\\ &=-9 \end{aligned}$ In conclusion, the average value of $-x^2+3$ on the interval $[0,6]$ is $-9$.